Every pupil should have the chance to learn from the neglected father of geometry
Towards the end of the 1960s, and during the 1970s, a decade when some people felt Britain was going to hell in a handcart, a group of ambitious intellectuals introduced the “new maths” in schools across the country. If traditional mathematics, numbers and equations were a bit hard for many people, why not replace them with something more up to date that everyone could understand? And as the rate of inflation went up and the winter of discontent set in, perhaps it was better to forget about numbers and let this bold cultural revolution in mathematics create a new generation of proletarian intellectuals able to take their place at the vanguard of a new understanding of mathematics. What happened?
At the time, my mother asked me, her son with a degree in mathematics: “Do you understand the new maths?” I was nonplussed. What new maths? Of course mathematics is developing new ideas and new methods all the time, but you can’t learn and understand them until you’ve learned the basics. Was this a new method of teaching the basics? Apparently not – they were learning about sets (very roughly speaking, collections of things), rather than numbers and algebra. When I recently asked a colleague why, he said it was meant to teach logic. Logic? Now that’s important. Logic at an elementary level deals with such things as syllogisms: all men are mortal; Socrates is a man; hence Socrates is mortal. And contrapositives: if rain implies clouds, then no clouds implies no rain. But it doesn’t imply that if there’s no rain, then there are no clouds. Common sense? Yes, but there is a logical structure here and the way to learn it is to relate it to things we understand, rather than abstractions that most people don’t understand terribly well.
Logical reasoning is important and it used to be part of a good mathematics course up to the age of 16. It was done under the heading of geometry, a venerable subject that has used careful rational argument for over two thousand years, ever since the Greek mathematician, Euclid of Alexandria, wrote the definitive text on the subject. Starting with simple axioms, such as the idea that between any two points there is a line, Euclid developed the main results of plane geometry. Assumptions were clearly stated, and the results, which we call theorems, were carefully proved so that no unspoken assumptions were made and no gaps appeared in the logic.
Not everyone did Euclidean geometry at school. A friend of mine who went to a secondary modern school and left at 15 only learned it later, but when he did he recalls that “I was bowled over by the fact that you could prove things, clearly and beyond doubt.” Had he passed the eleven-plus and gone to grammar school, he would certainly have taken a serious geometry course, but now you can go to school until you are 18, take A-levels and go to university without ever doing such a course.
What has replaced it? A bit of this and a bit of that, but the trouble with bits and pieces is that they don’t always hang together well, and things get learned by rote, and applied with calculator in hand.
This is rather strange because we are happy to inveigh against rote learning in third-world countries. We think people should learn to think for themselves, and find it regrettable when school fails to inculcate reasoned and rational argument.
What we need is rational thinking in terms of abstract concepts, and this is easiest when using abstractions we all understand, such as points and lines, rather than “sets” that often confuse students. I’ve known students at university being confused about the difference between the empty set, and the set consisting of zero, and even more abstractly the set consisting of the empty set, which is different again. Points and lines, on the other hand, seem real enough to most people, though they are in fact abstractions from reality – after all, lines in the material world have a certain thickness, and points a certain size. But as abstractions they are not hard to understand, and the arguments are logical, and visual, which helps. Moreover, Euclidean geometry, and more generally Greek rationalism, has a glorious history. It inspired early Islam, where scholars in Baghdad translated Euclid and other Greek authors into Arabic. These Arabic works were later translated into Latin and inspired new learning in medieval Europe. Later, during the Renaissance, Greek manuscripts were translated directly into Latin, and into the living languages of Europe. This allowed Euclid’s work to become the basis for teaching geometry, and learning geometry became synonymous with reading Euclid.
Do schools still teach Euclid? Yes, indeed they do, and my son did an excellent course on Euclidean geometry at high school in America. But sadly many British undergraduates in mathematics have never taken such a course, and are very unsteady on what exactly constitutes a proof. When they take an undergraduate degree, they have to get to grips with proofs in which the material is far more abstract than in Euclidean geometry. With the “new maths” they were supposed to be better at abstraction, but it doesn’t seem to work that way. The “new maths” was a failure, but before you say the answer is obvious, that we should get back to teaching the basics, be warned that this is easier said than done.
The problem with teaching any subject is that the teachers themselves have to understand it. When you throw something out of the usual curriculum, the teachers of the future don’t learn it, so it’s not easy for them to teach it. In a subject like English it’s not such a problem. A teacher who is unfamiliar with Shakespeare can always read a play and then do it in class, learning it better each time it is taught.
History, too, lends itself to this. But science and mathematics can be tough to learn well on your own, unless you’re specially talented. In fact, mathematics is the worst of all in this respect. Ideally, a teacher should understand what he or she is teaching and a lot more besides. Most mathematics teachers, grappling on the very edge of their knowledge, are not in a good position to take on things they were never taught themselves. This was the problem with the “new maths”, because most teachers didn’t understand it well or see the point.
Perhaps in the 1970s it didn’t seem to matter. If our children babbled about sets rather than numbers, why should we care? Who could overtake us? The Soviet Union? They had problems of their own, and even though their bright students got a fine education, they didn’t look as if they were about to overtake the commercial and industrial might of the West. But today, China and India have no qualms about teaching the basics, and making sure that teachers understand them.
As for those ambitious academics in Britain, and in Europe, who succeeded in promoting their brave new ideas, what happened to them? Nothing much. They got awards, even knighthoods, and retired. A pity about the pupils, though. They went through the new system and lost part of the heritage of mathematics. Some schools, particularly fee-paying schools, continued to teach it outside the usual exam curriculum, but many parents cannot afford this for their children. It should be available in all schools, for all pupils who can benefit.